Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation

Abstract

Burgers’ equation is used to describe wave phenomena in hydrodynamics and acoustics. It was derived originally as a prototype to provide analytic insight into the nature of turbulence and its modeling, and has found applications in the study of shock waves and wave transmission. Burgers’ equation is not globally controllable, and under certain conditions it can be neutrally stable. In this study, we explore the adaptive backstepping boundary control (BBC) methodology on a modified Burgers’ equation with unknown parameters, but constant, for the reactive and convective (nonlinear) terms, with Robin and Neumann boundary conditions (BCs), where this latter BC is actuated by the control signal. The nominal controller is designed from a linear partial differential equation (PDE), and under the assumption that this nominal controller also achieves stabilization for the modified Burgers’ equation, then its adaptive version is proposed for the control of such nonlinear PDE systems. Simulation results show convergence near the ideal values for the parametric estimates while the estimation error converges to zero.

Keywords

• adaptive control
• backstepping control
• boundary control
• infinite dimensional systems
• modified Burgers’ equation
• partial differential equations